1

Lecture 15: More about assignment and the Environment Model (EM)

2

Another thing the substitution model did not explain well:

• Nested definitions : block structure

(define (sqrt x) (define (good-enough? guess) (< (abs (- (square guess) x)) 0.001)) (define (improve guess) (average guess (/ x guess))) (define (sqrt-iter guess) (if (good-enough? guess) guess (sqrt-iter (improve guess)))) (sqrt-iter 1.0))

3

(sqrt 2) | GE

GE

p: xb:(define good-enou ..) (define improve ..) (define sqrt-iter ..) (sqrt-iter 1.0)

sqrt:

E1 x: 2good-enough:

p: guessb:(< (abs ….)

improve:sqrt-iter:

guess: 1

sqrt-iterguess: 1

good-enou?

4

Mutators for compound data

5

Compound Data

• constructor:

(cons x y) creates a new pair p

• selectors:

(car p) returns car part of pair

(cdr p) returns cdr part of pair

• mutators:

(set-car! p new-x) changes car pointer in pair

(set-cdr! p new-y) changes cdr pointer in pair

; Pair,anytype -> undef -- side-effect only!

6

Example: Pair/List Mutation

(define a (list 1 2))

(define b a)

a ==> (1 2)

b ==> (1 2) 1 2b

a

(set-car! a 10)

b ==> (10 2)

10

X

7

Example 2: Pair/List Mutation

(define x (list 'a 'b))

a b

xX

21

(set-car! (cdr x) (list 1 2))

1. Eval (cdr x) to get a pair object

2. Change car pointer of that pair object

• How mutate to achieve the result at right?

8

Sharing, Equivalence and Identity

• How can we tell if two things are equivalent?

-- Well, what do you mean by "equivalent"?

1. The same object: test with eq?(eq? a b) ==> #t

2. Objects that "look" the same: test with equal?(equal? (list 1 2) (list 1 2)) ==> #t(eq? (list 1 2) (list 1 2)) ==> #f

• If we change an object, is it the same object? -- Yes, if we retain the same pointer to the object

• How tell if parts of an object is shared with another? -- If we mutate one, see if the other also changes

9

x ==> (3 4)

y ==> (1 2)

(set-car! x y)

x ==>

followed by

(set-cdr! y (cdr x))

x ==>

Your Turn

3 4

x

1 2

y

((1 2) 4)

((1 4) 4)

10

x ==> (3 4)

y ==> (1 2)

(set-car! x y)

x ==>

followed by

(set-cdr! y (cdr x))

x ==>

Your Turn

3 4

x

1 2

y

((1 2) 4)

X

((1 4) 4)

X

11

Summary

• Scheme provides built-in mutators•set! to change a binding•set-car! and set-cdr! to change a pair

• Mutation introduces substantial complexity• Unexpected side effects• Substitution model is no longer sufficient to explain

behavior

12

Stack and queues

13

Stack Data Abstraction• constructor:

(make-stack) returns an empty stack

• selectors: (top stack) returns current top element from a stack

• operations: (insert stack elt) returns a new stack with the element

added to the top of the stack

(delete stack) returns a new stack with the topelement removed from the stack

(empty-stack? stack) returns #t if no elements, #f otherwise

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Stack Data Abstraction

Top

Insert

15

Stack Data Abstraction

Top

Insert

16

Stack Data Abstraction

Top

Insert

17

Stack Data Abstraction

Top

Delete

19

Stack Implementation Strategy

• implement a stack as a list

dba

• we will insert and delete items off the front of the stack

20

Stack Implementation

(define (make-stack) nil)

(define (empty-stack? stack) (null? stack))

(define (insert stack elt) (cons elt stack))

(define (delete stack) (if (empty-stack? stack) (error "stack underflow – delete") (cdr stack)))

(define (top stack) (if (empty-stack? stack) (error "stack underflow – top") (car stack)))

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Limitations in our Stack

• Stack does not have identity

(define s (make-stack))

s ==> ()

(insert s 'a) ==> (a)

s ==> ()

(set! s (insert s 'b))

s ==> (b)

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• Attach a fixed pair. This provides an object whose identity remains even as the object mutates !

• Can also attach a type tag – defensive programming

Alternative Stack Implementation – pg. 1

• Note: This is a change to the abstraction! User should know if the object mutates or not in order to use the abstraction correctly.

acdstack

s X

(delete! s)

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Alternative Stack Implementation – pg. 2

(define (make-stack) (cons 'stack nil))

(define (stack? stack)

(and (pair? stack) (eq? 'stack (car stack))))

(define (empty-stack? stack)

(if (not (stack? stack))

(error "object not a stack:" stack)

(null? (cdr stack))))

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Alternative Stack Implementation – pg. 3

(define (insert! stack elt) (cond ((not (stack? stack)) (error "object not a stack:" stack)) (else (set-cdr! stack (cons elt (cdr stack))) stack)))

(define (delete! stack) (if (empty-stack? stack) (error "stack underflow – delete") (set-cdr! stack (cddr stack))) stack)

(define (top stack) (if (empty-stack? stack) (error "stack underflow – top") (cadr stack)))

25

Queue Data Abstraction (Non-Mutating)

• constructor: (make-queue) returns an empty queue

• accessors: (front-queue q) returns the object at the front of the

queue. If queue is empty signals error• mutators:

(insert-queue q elt) returns a new queue with elt at the rear of the queue

(delete-queue q) returns a new queue with the item at the

front of the queue removed • operations:

(empty-queue? q) tests if the queue is empty

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Queue illustration

x1fr

ont

Insert

27

Queue illustration

x1fr

ont

Insert

x2

28

Queue illustration

x1fr

ont

Insert

x2 x3

29

Queue illustration

fron

t

Delete

x2 x3

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Queue Contract

• If q is a queue, created by (make-queue) and subsequent queue procedures, where i is the number of insertions, j is the number of deletions, and xi is the ith item inserted into q , then

• If j>i then it is an error

• If j=i then (empty-queue? q) is true, and (front-queue q) and

(delete-queue q) are errors.

• If j<i then (front-queue q) = xj+1

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Simple Queue Implementation – pg. 1

• Let the queue simply be a list of queue elements:

c db

• The front of the queue is the first element in the list• To insert an element at the tail of the queue, we need to

“copy” the existing queue onto the front of the new element:

d newcb

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Simple Queue Implementation – pg. 2

(define (make-queue) nil)

(define (empty-queue? q) (null? q))

(define (front-queue q) (if (empty-queue? q) (error "front of empty queue:" q) (car q)))

(define (delete-queue q) (if (empty-queue? q) (error "delete of empty queue:" q) (cdr q)))

(define (insert-queue q elt) (if (empty-queue? q) (cons elt nil) (cons (car q) (insert-queue (cdr q) elt))))

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Simple Queue - Orders of Growth

• How efficient is the simple queue implementation?• For a queue of length n

– Time required -- number of cons, car, cdr calls?– Space required -- number of new cons cells?

• front-queue, delete-queue:

• Time: T(n) = O(1) that is, constant in time• Space: S(n) = O(1) that is, constant in space

• insert-queue:• Time: T(n) = O(n) that is, linear in time• Space: S(n) = O(n) that is, linear in space

34

Queue Data Abstraction (Mutating)

• constructor: (make-queue) returns an empty queue

• accessors: (front-queue q) returns the object at the front of the

queue. If queue is empty signals error• mutators:

(insert-queue! q elt) inserts the elt at the rear of the queueand returns the modified queue

(delete-queue! q) removes the elt at the front of the queueand returns the modified queue

• operations:

(queue? q) tests if the object is a queue

(empty-queue? q) tests if the queue is empty

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Better Queue Implementation – pg. 1

• We’ll attach a type tag as a defensive measure• Maintain queue identity• Build a structure to hold:

• a list of items in the queue• a pointer to the front of the queue• a pointer to the rear of the queue

queue

c dba

front-ptr

rear-ptr

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Queue Helper Procedures• Hidden inside the abstraction

(define (front-ptr q) (cadr q))

(define (rear-ptr q) (cddr q))

(define (set-front-ptr! q item)

(set-car! (cdr q) item))

(define (set-rear-ptr! q item)

(set-cdr! (cdr q) item))

queue

c dba

front-ptr

rear-ptr

37

Better Queue Implementation – pg. 2

(define (make-queue) (cons 'queue (cons nil nil)))

(define (queue? q) (and (pair? q) (eq? 'queue (car q))))

(define (empty-queue? q) (if (not (queue? q)) ;defensive (error "object not a queue:" q) ;programming (null? (front-ptr q))))

(define (front-queue q) (if (empty-queue? q) (error "front of empty queue:" q) (car (front-ptr q))))

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Queue Implementation – pg. 3

(define (insert-queue! q elt)

(let ((new-pair (cons elt nil)))

(cond ((empty-queue? q)

(set-front-ptr! q new-pair)

(set-rear-ptr! q new-pair)

q)

(else

(set-cdr! (rear-ptr q) new-pair)

(set-rear-ptr! q new-pair)

q))))

queue

c dba

front-ptr

rear-ptr

e

39

Queue Implementation – pg. 4

(define (delete-queue! q)

(cond ((empty-queue? q)

(error "delete of empty queue:" q))

(else

(set-front-ptr! q

(cdr (front-ptr q)))

q)))

queue

c dba

front-ptr

rear-ptr

40

Time and Space complexities ?

• O(1)